Appl. Math. Mech. -Engl. Ed., 32(9), 1197–1206 (2011) DOI 10.1007/s10483-011-1493-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011
Applied Mathematics and Mechanics (English Edition)
Global exponential stability of switched systems∗ V. FILIPOVIC (Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, Kraljevo 36000, Serbia)
Abstract This paper proposes a method for the stability analysis of deterministic switched systems. Two motivational examples are introduced (nonholonomic system and constrained pendulum). The finite collection of models consists of nonlinear models, and a switching sequence is arbitrary. It is supposed that there is no jump in the state at switching instants, and there is no Zeno behavior, i.e., there is a finite number of switches on every bounded interval. For the analysis of deterministic switched systems, the multiple Lyapunov functions are used, and the global exponential stability is proved. The exponentially stable equilibrium of systems is relevant for practice because such systems are robust to perturbations. Key words
switched system, multiple Lyapunov function, global exponential stability
Chinese Library Classification O322 2010 Mathematics Subject Classification
1
70K20
Introduction
Hybrid systems describe the interaction between software, modeled by finite state-systems such as finite-state machines, and the physical world described by differential equations. The several key verification and control synthesis results for hybrid systems, guided by the concept of bisimulation, are outlined in Ref. [1]. From the classical control theory point of view, the hybrid systems can be described as a switching control between analog feedback loops[2–3] . For example, the systems with quantization are the examples of switched systems with state dependent switching. The main motivation for studying such kinds of systems is a fact that there exist a large class of nonlinear systems, which is possible to be stabilized by switching control schemes, but cannot be stabilized by any continuous static state feedback control[4] . The overview of the most recent developments in the field of the stability and stabilizability of switched systems is presented in Refs. [5] and [6]. In Ref. [7], a framework was established for the stability analysis of deterministic and stochastic switched systems by combining the method of multiple Lyapunov functions with the comparison principle. It is shown that nonlinear deterministic switched systems are globally uniformly asymptotically stable. Reference [8] considers the stability of deterministic linear switched systems. The problem is based on the determination of a minimum dwell time by means of a ∗ Received Feb. 9, 2011 / Revised May 1, 2011 Project supported by the Ministry of Science and Technological Development of the Republic of Serbia (No. TR-3326) Corresponding author V. FILIPOVIC, Professor, E-mail:
[email protected]
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family of quadratic Lyapunov functions. An important detail of my result is that it includes the stability of possible sliding modes. In Ref. [9], the stabilization of switched linear control systems with a static linear feedback was considered. It was considered a stronger version of stabilization, namely, stabilization maintained when the feedback gain increased. Such approach is valid for a large class of switched systems. In Ref. [10], the comparison between a switching controller and two linear time-invariant (LTI) controllers for a class of LTI plants was considered. It was shown that the switching architecture could outperform the first-order LTI control. The result is important for practice. Namely, using philosophy of the switching control is possible to design controllers with fixed structures for high order models for the real processes. Reference [11] considers the switching control of time-delay systems. The analog part of the system was described by finite set of continuous-time models with input delays, and the unmodeled dynamics was in the form of the affine matrix family. As a result, the robust LQ switching controllers were given. In Ref. [12], it was considered the application of switching controllers when in the system actuator saturation was presented. That is the most frequent nonlinearity encountered in the practice. Reference [13] describes the application of switching system in the air traffic management. In this paper, we consider the stability of switched deterministic systems. For system analysis, we use multiple Lyapunov functions[14] . It is assumed as follows: a) there is no jump in the state x at the switching instants, b) there is no Zeno behavior, i.e., there are a finite number of switches on every bounded interval of time. The situation with jumps in the state of the system is considered in Refs. [15] and [16]. Here, we consider the exponential stability of switched systems. The finite set of models consists of nonlinear models. The exponentially stable equilibrium is relevant for practice. Namely, such systems are robust to perturbations.
2
Two examples
In this part, we will describe two examples. The first one is the nonholonomic system, which is not stabilizable by means of any individual continuous state feedback controller. For these systems, multicontroller switching provides a good conceptual framework to solve the problem. The second example (constrained pendulum) describes the multi-model systems (two models). Example 1 Consider the system[2] ⎧ n ⎪ ⎨ x(t) ˙ = gi (x(t))ui (t) = G(x(t))u(t), ⎪ ⎩
i=1 n
x(t) ∈ R , u(t) ∈ R , G(x(t)) ∈ R m
n×m
(1) ,
which is known as a nonholonomic system. It means that the system is subject to constraints involving both the state x(t) (position) and its derivative x(t) ˙ (velocity). The stability of equilibrium of nonholonomic systems with nonlinear constraints was considered in Ref. [17]. In Ref. [18], the problem of the formation control for the multiple nonholonomic agents on a plane was solved. If rank G(0) = m and m < n, the system (1) violates Brockett’s condition[19] . We have the next result. Result 1 A continuous feedback law cannot asymptotically stabilize the system (1) with rank G(0) = m < n . A simple example of the nonholonomic system will be considered as the wheeled mobile robot (unicycle) (see Fig. 1)[19] .
Global exponential stability of switched systems
Fig. 1
1199
Unicycle
The state variables are x1 (t), x2 (t), and θ(t), and the kinematics of the robot have the following form: ⎧ ⎪ ⎨ x˙ 1 (t) = u1 (t) cos(θ(t)), x˙ 2 (t) = u1 (t) sin(θ(t)), (2) ⎪ ⎩˙ θ(t) = u2 (t), where u1 (t) and u2 (t) are the control inputs (the forward and the angular velocities, respectively). Asymptotic stabilization of the system means parking the unicycle at the origin and aligning it with the x1 -axis. The system (2) takes the form (1) with n = 3 and m = 2 as follows: ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ x1 (t) cos(θ(t)) 0 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ x(t) = ⎣ x2 (t)⎦ , g1 (x(t)) = ⎣ sin(θ(t))⎦ , g2 (x(t)) = ⎣ 0⎦ . (3) θ(t) 0 1 From Result 1, it follows that the unicycle problem cannot be solved by means of continuous feedback. Example 2 Consider a mathematical pendulum with the length l that hits a pin such that the constrained pendulum has a length lc (see Fig. 2). The continuous state space variable is x = (θ, ω), where ω is the angular velocity of the end of the pendulum. We have a hybrid system with two locations (unconstrained and constrained).
Fig. 2
Constrained pendulum
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The unconstrained dynamics, valid for θ θpin , described in the next form[20] : ⎧ ⎪ ˙ = 1 ω(t), ⎨ θ(t) l ⎪ ⎩ ω(t) ˙ = −g sin(θ(t)) − αω(t),
(4)
where α is the friction coeficient, and g is the acceleration. The constrained dynamics valided for θ > θpin is given by the same equations with l replaced by lc . We see that there is only discontinuity in the right hand side of the differential equation, which is caused by the change from l to lc . The complete behavior of the constrained pendulum is described with two models.
3
Problem formulation
Let us suppose that E1 and E2 are subsets of the Euclidean space. Let C[E1 , E2 ] denote the space of all continuous function f : E1 → E2 and C 1 [E1 , E2 ] denote the space of all functions f : E1 → E2 that are once continuously differentiable. The functions α ∈ C[R0 , R0 ] is of class K if α is increasing and α(0) = 0. For more details, see Ref. [18]. The autonomous switched nonlinear system can be modeled as x(t) ˙ = fp (x(t)),
t ∈ R+ ,
∀p ∈ P = {1, 2, · · · , N },
(5)
where fp = R n → R n ,
f (0) = 0,
∀p ∈ P,
(6)
the state x ∈ Rn , and R+ denote the non-negative real numbers. The finite set P is an index set and stands for the collection of discrete modes. The switching signal is a logical rule that orchestrates switching between subsystems. That is classes of piecewise constant maps. σ : [0, ∞] → P.
(7)
Such a function has a finite number of discontinuities on every bounded time interval and takes constant values on every interval between two consecutive switching times. By requesting a switching signal be piecewise constant, we mean that the switching signal σ(t) has finite number of discontinuities on any finite interval from R+ . This corresponds to no-chattering requirement for the continuous time switched systems. The logical rule that generates the switching signal constitutes the switching logic and the index p = σ(t) is the active mode at the time instant t. The active mode at the time t may depend on the time instant t, current state x(t), and previous active mode σ(τ ) for τ < t. In this paper, we will consider the switching logic, which depends on time t only. σ is continuous from the right everywhere, σ(t) = lim+ σ(τ ), τ →t
∀τ 0.
(8)
The switched system for the family (6) generated by σ is x(t) ˙ = fσ(t) (x(t)).
(9)
Figure 3 represents the switching system with the switching signal. Although the continuous part of the system is autonomous, the results of the paper are important for stabilization problems (design of controllers). Now, we will define the notion of the global exponential stability.
Global exponential stability of switched systems
Fig. 3
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Switching control systems
Definition 1 The equilibrium of system (6) is globally exponentially stable if there exist constants (k1 , k2 ) > 0 such that x(t) k1 x(t0 ) exp{−k2 (t − t0 )} for ∀t, to 0 and ∀x(to ) ∈ Rn , where · is Euclidean norm. For our main result, we need the following lemma, which is adopted from Ref. [19]. Lemma 1 Consider the system having index p in the family (6). Suppose that there exist constants (a, b, c, r) > 0 , m 1, and a C 1 [Rn , R0 ] function Vp : Rn → Rn such that m m (i) a x(t) Vp (x(t)) b x(t) , ∀t > 0, ∀x(t) ∈ Rn . m (ii) V˙ p (x(t)) −c x(t) , ∀t 0, ∀x(t) ∈ Rn . Then, the equilibrium of (6) is globally exponentially stable. Clearly, exponential stability is a stronger property than uniform asymptotic stability. In addition, exponential stability follows asymptotic stability. However, globally exponentially stable equilibrium of the system is robust to perturbations.
4
Exponential stability of switched systems
Now, we will formulate the main result of this paper. Theorem 1 Consider the switched system (9). Let us suppose that the following assumptions are valid : (A1 ) Index set P is finite, i.e., P = {1, 2, · · · , N }. (A2 ) Function Vp (·) ∈ C 1 [Rn , R0 ] for ∀p ∈ P . (A3 ) Function U (·) ∈ K. (A4 ) For ∀x(t) ∈ Rn , m 1, ∀p ∈ P and a, b > 0, a < b, a x(t)
m
m
Vp (x(t)) b x(t) .
(A5 ) For ∀x(t) ∈ Rn , m 1, ∀p ∈ P and c > 0, m V˙ p (x(t)) −c x(t) .
(A6 ) For every p ∈ P and every pair of switching instants σ(ti ) = σ(tj ) = p,
∀(ti , tj ),
i 0 and restrict the class of admissible switching signals so that for switching times t1 , t2 , · · · satisfy the inequality t1+1 − ti τd ,
∀i.
(26)
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The number τd is known as the dwell time. When the nonlinear systems in the family (6) are exponentially stable, the switched system is exponentially stable if the dwell time τd is sufficiently large. In this example, we will find a lower bound for τd using the result of Theorem 1. Let us consider the simple case when P = {1, 2} and σ takes 1 on [t0 , t1 ) and 2 on [t1 , t2 ), where t1+1 − ti τd , i = 0, 1 (27) as shown in Fig. 4.
Fig. 4
Dwell time switching pattern
From the relations (18) and (27) and (A4 ) of Theorem 1, we have
b 2
c Vp (x(t2 )) exp − (t2 − t0 ) Vp (x(t0 )) a b
b 2
2c exp − τd Vp (x(t0 )). a b
(28)
From (A6 ) of Theorem 1, we have Vp (x(t2 )) − Vp (x(t0 )) −γ x(t0 )2 ,
∃γ > 0.
By using (28) and (29), it follows that
b 2
2c 2 exp − τd − 1 Vp (x(t0 )) −γ x(t0 ) . a b Relation (30) together with A4 of Theorem 1 gives
b 2
2c exp − τd − 1 b −γ. a b Since γ can be arbitrary positive number from relation (31), we have
b 2
2c exp − τd 1. a b From (32), we finally have
(29)
(30)
(31)
(32)
b b ln . (33) c a This is a desired lower bound on the dwell time. Example 4 Stability of switched systems The stability issues of switched systems include several interesting phenomena. For example, even when all subsystems are exponentially stable, the switched systems, for certain switching signals may be unstable[22] . In addition, one may carefully switch between unstable subsystems to make the switched system exponentially stable[22] . The important problem is the stability analysis of the switched systems under a given switching signal (arbitrary, slow switching, etc.). In that case, Theorem 1 is directly applicable. τd
Global exponential stability of switched systems
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Conclusions
In this paper, the analysis of the exponential stability of the switched systems is considered. The multiple models are given in the form of autonomous nonlinear differential equations. The analysis is based on the multiple Lyapunov functions, and the global exponential stability is proved. The exponentially stable systems are robust with respect to unmodeled dynamics. Some applications of Theorem 1 are presented (the stability of switched systems with dwell time, the stability of switched systems for given switching signal). The results will be extended to the nonautonomous continuous systems (controller design). In addition, it is interesting to extend the results to input–to–state stability of switched systems[18] .
References [1] Tabuda, P. Verification and Control of Hybrid Systems. A Symbolic Approach, Springer, Berlin (2009) [2] Liberzon, D. Switching in Systems and Control, Springer, Berlin (2003) [3] Sun, Z. and Ge, S. Switched Linear Systems: Control and Design, Springer, Berlin (2005) [4] Brockett, R. W. Asymptotic stability and feedback stabilization. Differential Geometric Control Theory, Birkhauser, Boston, 181–191 (1983) [5] Cortes, J. Discontinuous dynamical systems. IEEE Control Systems Magazine, 25(1), 36–73 (2008) [6] Lin, H. and Antsaklis, P. J. Stability and stabilizability of switched linear systems: a suvey of recent results. IEEE Transactions on Automatic Control, 54(2), 308–322 (2009) [7] Chatterjee, D. and Liberzon, D. Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov function. SIAM Journal on Control and Optimization, 45(1), 174–206 (2006) [8] Geromel, J. C. and Colaneri, P. Stability and stabilization of continuous-time switched linear systems. SIAM Journal on Control and Optimization, 45(5), 1915–1930 (2006) [9] Artstein, Z. and Romen, J. On stabilization of switched linear systems. Systems and Control Letters, 57(11), 919–926 (2008) [10] Santarelli, K. R. and Dahleh, M. A. Comparison between a switching controller and two LTI controllers for a class of LTI plants. International Journal of Robust and Nonlinear Control, 19(2), 185–217 (2009) [11] Filipovic, V. Z. Hybrid control of systems with input delay. IFAC World Congress, Praha, Czech Republic (2005) [12] Filipovic, V. Z. Switching control in the presence of constraints and unmodeled dynamics. New Approaches in Automation and Robotics, InTech, Vienna, 227–238 (2008) [13] Filipovic, V. Z. and Nedic, N. Stability if switched stochastic nonlinear systems. Air Traffic Control, SCYO, Vienna, 23–38 (2010) [14] Branicky, M. S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 43(4), 475–482 (1998) [15] Guan, Z., Hill, D., and Shen, X. On hybrid impulsive and switching systems and application to nonlinear control. IEEE Transactions on Automatic Control, 50(7), 1058–1062 (2005) [16] Li, Z., Soh, Y., and Wen, C. Switched and Impulsive Systems, Springer, Berlin (2005) [17] Covic, V., Veskovic, M., Djuric, D., and Obradovic, A. On the stability of equilibria of nonholonomic systems with nonlinear constraints. Appl. Math. Mech.-Engl. Ed., 31(6), 751–760 (2010) DOI 10.1007/s10483-010-1309-7 [18] Wu, F. and Geng, Z. Formation control for nonholonomic agents uising passivity techniques. Appl. Math. Mech.-Engl. Ed., 31(1), 27–36 (2010) DOI 10.1007/s10483-010-0104-x
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[19] Sastry, S. Nonlinear Systems: Analysis, Stability and Control, Springer, Berlin (1999) [20] Van der Schaft, A. and Schumacher, H. An Introduction to Hybrid Dinamical Systems, Springer, Berlin (2000) [21] Morse, A. S. Supervisory control of families of linear set-point controllers, part 1. exact matching. IEEE Transactions on Automatic Control, 41(10), 1413–1431 (1996) [22] Decarlo, R. A., Branicky, M. S., Pettersson, S., and Lennartson, B. Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7), 1069–1082 (2000) [23] Isidori, A. Nonlinear Control Systems II, Springer, Berlin (1999) [24] Vidyasagar, M. Nonlinear Systems Analysis, SIAM, Philadelphia (2002)
